Humans can recognize objects in a way that is invariant to scale, translation, and clutter. We use invariance theory as a conceptual basis, to computationally model this phenomenon. This theory discusses the role of eccentricity in human visual processing, and is a generalization of feedforward convolutional neural networks (CNNs). Our model explains some key psychophysical observations relating to invariant perception, while maintaining important similarities with biological neural architectures. To our knowledge, this work is the first to unify explanations of all three types of invariance, all while leveraging the power and neurological grounding of CNNs.